(1)   if the control domain is convex, \(f_1=f_2=f_3\equiv 0\) , \(\sigma _3\equiv 1\) , \(b_2\)  is independent of \((y,z^1,z^2,\tilde{z}^1,\tilde{z}^2,u)\) , \(l,g\)  are independent of \((\tilde{z}^1,\tilde{z}^2)\)  with \(\tilde{z}^1=\tilde{z}^2\equiv 0\) , Theorem REF  reduces to Theorem 2.1 of Wang et al. [1]}, the local maximum principle of partially observed case for FBSDEs with correlated Brownian noises with the following important relations:
\(\begin{aligned}&\tilde{\Gamma }_t=Z_t,\ b_2(\cdot ,\cdot )=h(\cdot ,\cdot ),\ h_t=-p_tZ_t,\\&(r_t,s^1_t,s^2_t)=(P_t,Q_t,\tilde{Q}_t),\\&m_t=q_tZ_t,\ n^1_t=k_tZ_t,\ n^2_t=h(t,\bar{x})q_tZ_t+\tilde{k}_tZ_t,\end{aligned}\) 
